Basic results Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol37.Issue1.Jan1999:

L . Arvan et al. Mathematical Social Sciences 37 1999 97 –106 99

2. Basic results

Consider the following two-player game in normal form, denoted by G: S is the space i of player i’s pure strategies and s is a generic element of S , i 5 1, 2; S ; S 3 S and i i 1 2 s ; s , s ; finally, u s denotes player i’s utility. 1 2 i Based on game G, we now define an extended game, denoted by G, which involves a communication stage prior to playing G. At the communication stage, players simul- taneously announce actions from their own pure-strategy sets S the terms ‘action’ and i ‘pure strategy’ meaning the same thing. After all messages are sent and received, game G is played. Games of this sort always admit equilibria in which cheap talk has no influence on the 4 outcome of G ‘babbling equilibria’. Following Farrell 1987, we concentrate on no-babbling equilibria, that is, equilibria with the following properties: 1. If announcements correspond to a Nash equilibrium of G, then that equilibrium becomes focal and is thus played. 2. If announcements do not correspond to a Nash equilibrium of G, then G is played as if no communication has taken place: a ‘default’ equilibrium is played independently of which announcements are made. Payoffs in this default equilibrium are given by ¯u . i Notice that properties i–ii induce a well-defined reduced game in the communica- tion stage, so the set of no-babbling equilibria Nash, perfect Nash, or proper Nash is nonempty. Finally, we define s to be a superior equilibrium of G if and only if s is a ¯ pure-strategy equilibrium and u s . u , ;i. Our main result will be based on the i i hypothesis that there exists at least one superior equilibrium. This hypothesis implies that G is to some extent a game of coordination, specifically, a game with at least two Nash equilibria which are Pareto ranked. Theorem 1. Assume that all pure-strategy Nash equilibria of G are strict . If there exists some superior equilibrium of the original game , then, in a proper no-babbling equilibrium of the extended game , only actions corresponding to superior equilibria are announced , and the expected payoff of both players is strictly greater than in the game with no communication . Proof. Consider a ‘perturbed’ game in the communication stage. In this game, every message is announced with positive probability. Therefore, it is a strictly dominated strategy for player i to announce an action corresponding to a pure-strategy Nash 4 See, for example, Farrell and Gibbons 1989. Seidmann 1992 shows that the same is not necessarily true in games of incomplete information. 100 L . Arvan et al. Mathematical Social Sciences 37 1999 97 –106 ¯ equilibrium, say s, yielding him or her a payoff less than u . In fact, the expected i ¯ payoff of announcing such an action would be lower than u , by the assumption that all i pure-strategy Nash equilibria are strict; whereas announcing an action corresponding to a ¯ superior equilibrium, say s, would yield player i an expected payoff greater than u . i By the requirement of properness Myerson, 1978, the weight assigned by player i to the action corresponding to s must be at least one order of magnitude lower than the weight assigned to the action corresponding to s. Now, given the above, it is also a dominated strategy for player j to announce the action corresponding to s, assuming that payoffs are bounded and that the ‘perturbed’ game is sufficiently close to the ‘unperturbed game.’ To see why, notice that player j’s expected payoff from announcing the action associated to s is ¯ A 5 Ps u s 1 1 2 Ps u , 1 i j i j where Ps is the probability that action s is announced by player i. On the other i i hand, announcing the action corresponding to s yields player j an expected payoff of ¯ B 5 Ps u s 1 1 2 Ps u . 2 i j i j ¯ Since u s.u and Ps ePs the latter by properness, we have A,B. j j i i Finally, a similar argument, if somewhat simpler, applies to actions corresponding to no pure-strategy Nash equilibrium. Likewise, it is straightforward to show, by contradic- tion, that at least one superior equilibrium must be played with positive probability. j It should be remarked that the argument extends to games with T .1 rounds of cheap talk a case also considered in Farrell 1987. It suffices to note that the proof applies to any period t ,T, with the difference that payoffs in case of no agreement at stage t are t t t ¯ ¯ ¯ ¯ now given by u , u , where u .u . 1 2 i i

3. Counterexamples